Lindelöf Hypothesis

Let  be the least upper bound of the numbers  such that  is bounded as , where  is the Riemann zeta function. Then the Lindelöf hypothesis states that  is the simplest function that is zero for  and  for .

The Lindelöf hypothesis is equivalent to the hypothesis that  (Edwards 2001, p. 186).

Backlund (1918-1919) proved that the Lindelöf hypothesis is equivalent to the statement that for every , the number of roots in the rectangle  grows less rapidly than  as  (Edwards 2001, p. 188).

REFERENCES:

Backlund, R. "Über die Beziehung zwischen Anwachsen und Nullstellen der Zeta-Funktion." Ofversigt Finka Vetensk. Soc. 61, No. 9, 1918-1919.

Edwards, H. M. Riemann's Zeta Function. New York: Dover, 2001.

Lindelöf, E. "Quelque remarques sur la croissance de la fonction ." Bull. Sci. Math. 32, 341-356, 1908.